Factual error: The German agent pretending to be an American pilot is wearing a one piece flying suit from the 1960's when he is questioned by Hogan and the British escapee in the barracks. It even has the US Air Force logo on the sleeve, which didn't come into use until 1947, after the war.
Factual error: When the crew in the plane gets ready to drop the bombs on the refinery, Hogan (in the pilot seat) tells Carter (who is looking through the bomb sight) exactly when to push the button. In fact, the pilot of a WW2 bomber had no way of even knowing when exactly to release the bombs - it was the job of the bombardier, looking through his targeting optics, to know that. Carter at the bomb sight should be telling Hogan how exactly to steer to get the bombs onto the target.
Factual error: In the ammunition dump, a sign saying "Warnung Hoche Sprenggefahr" can be seen. That is not correct German. First, and foremost, it must be "hohe" and not "hoche", the latter form does not exist. Secondly, the word "Sprenggefahr", while not formally wrong, was never used in German. Depending on what the meaning of the sign is supposed to be, it must either be "Explosionsgefahr", if the overall danger of an explosion is meant, or if the property of the ammunition of being explosive is meant, it should be "Hochexplosiv." If the general presence of explosive material is being warned of, "Explosivstoffe" or "Hochexplosives Material" would be possible.
Plot hole: The boys take out the radio control unit from the mini tank. Le Beau is needed to get in and drive it to make it appear okay. After going between the buildings, they get him out, and start it on its way. Trouble is, no remote control and no driver, yet it turns and operates smoothly.
Character mistake: As Stauffen parts company with Col. Hogan at the road checkpoint, he salutes Hogan. While that is understandable under the circumstances - after all Hogan just saved his life, it would also be highly suspicious. Both Schultz and Stauffen's adjutant were watching them, so it can't be said they did it surreptitiously either. After all, Stauffen is a German general whereas Hogan is an allied prisoner officer of inferior rank. For an officer of higher rank to initiate the salute is a demonstration of great respect and/or thanks, which, as far as anyone besides Stauffen and Hogan knew, was not warranted in this situation.
Continuity mistake: Hogan Gives a Birthday Party when the prisoners capture a German plane. When it is taking off, it is an English Lancaster bomber with a twin tail. As it is flying overhead, it is an American B-17 with a single tail. Finally as it is shown flying, it is a German FW-200 maritime bomber.
Deliberate mistake: The film crew did not always bother to switch out the rank insignia on the uniforms of extras. In this episode, that leads to all kinds of enlisted and noncom rank insignia being present in the lineup of brand-new recruits when Klink inspects them. Rightfully they should all wear a single eagle on a borderless patch of a Flieger on their collar.
Factual error: In Klink's quarters, there's an (obviously fake) banana in the fruit bowl. Bananas were luxury items in Europe in the 1940s, and certainly so in war-torn Germany. The majority of Germans didn't have access to exotic fruit until the 1960s, and fake fruit of this type wasn't available then either.